Question #85af2

In terms of actually understanding, I find it is easier to look at the reverse process first, namely differentiation. We know for example that, for some constant #alpha#:

#d/dx (e^(alphax) ) = alpha e^(alphax) qquad triangle#

(Incidentally, this follows from the more general conclusion that: #d/dx (e^(f(x)) ) = f'(x) e^(f(x))#,..., which you can reach very quickly by using logarithmic differentiation.)

It follows from integrating #triangle# that:

#color(red) (int d/dx (e^(alpha x) ) \ dx) =int alpha e^(alpha x) \dx#

Or, by realising that in the red term the integration and differentiation "cancel out" (by the Fundamental Theorem of Calculus), and then reversing the order:

#int alpha e^(alpha x) \dx = e^(alpha x) +C#

Or, moving the constant across:
# int e^(alpha x) \dx = 1/alpha e^(alpha x) + C#

Then, adding a new constant #beta#, to reflect your problem:
# int beta e^(alpha x) \dx = beta/alpha e^(alpha x) + C#

So with #alpha = 0.1#, #beta = 2#:

# int 2 e^(0.1 x) \dx = 2/0.1 e^(0.1 x) + C#
#= 20 e^(0.1 x) +C #

That's all rather mechanical and somewhat dry, but I think it might help find a way of doing these that doesn't involve much memory.